Date: Jan 16, 2013 3:34 AM
Author: Jussi Piitulainen
Subject: Re: simplifying rational expressions

stony writes:

> Need a little help with this. We are simplifying the following, but
> the solution is pretty lengthy and messy because of the enormous
> number of factors. I was thinking that may be I am missing seeing a
> pattern (some series or something). Is grunt work the only way to
> solve this or is there a pattern that can simplify the whole process?
> My daughter was trying to solve this, but ended up with the mess and
> then I got the same mess, but I thought there may be an easy way to
> simplify this that I may be missing.
> ((b-c)/((a-b)(a-c))) + ((c-a)/((b-c)(b-a))) + ((a-b)/((c-a)(c-b))) +
> (2/(b-a)) - (2/(c-a))
> of course, I took all the factors in the denominator and then started
> multiplying the numerator with the remaining factors to end up with a
> mess.

I suspect you are missing the fact that (a - b) and (b - a) in the
denominators are essentially the same factor. The common denominator
of the terms is (a - b)(a - c)(b - c). I think you can leave the
denominator in that form.

I did the numerator two ways: in terms of these same factors, and
multiplying out and combining terms. They seemed about equally simple
to me, though I may have made mistakes. I usually do. Be careful with
the signs :)

Hope this helps.